Instantaneous phasor determination for poly-phase electrical grids

ABSTRACT

In one embodiment, three substantially simultaneous phase waveforms may be converted into a first quadrature signal and a zero sequence signal. For each phase waveform, a power system digital frequency may be determined through analysis of the first quadrature signal (e.g., and at least one additional prior quadrature signal) while eliminating waveform phase angles from the analysis. Subsequently, demodulation of the first quadrature signal and zero sequence signal based on the power system digital frequency results in a positive sequence phasor, a negative sequence phasor, and a zero sequence phasor.

TECHNICAL FIELD

The present disclosure relates generally to electrical grids, and, moreparticularly, to determining phasors in electrical grids.

BACKGROUND

Electric utilities use alternating-current (AC) power systemsextensively in generation, transmission, and distribution. Most of thesystems and devices involved operate on three-phase power, wherevoltages and currents are grouped in threes, with the waveformsstaggered evenly. The basic mathematical object that describes an ACpower system waveform (current of voltage) is the “phasor” (phase anglevector).

Devices known as Phasor Measurement Units (PMUs) have beencommercialized by several companies to calculate phasors from powerwaveforms. Because phase angle is a relative quantity, it is necessarywhen combining phasors taken from different parts of a power grid toalign the phase angle elements to a common phase reference; this is donein PMUs through the use of GPS timing signals. Such phasors are known assynchrophasors. PMUs measure synchrophasors, but existing devices usephasor calculation methods that have a number of shortcomings. Forexample, in addition to being generally expensive and burdensome toembedded processors (and thus mostly deployed in transmission gridsrather than distribution grids), PMUs generally only calculate thepositive sequence, even for unbalanced phasors, but all three sequencesare needed for many applications. Furthermore, PMUs use batch-typecalculations to produce phasor values, and are thus inadequate toprovide phasors on a sub-cycle basis, such as for various protectionapplications. Moreover, today's synchrophasor calculation methodsgenerally introduce various errors, such as phase lag and group delay,and do not account for variance in actual power system frequency.

BRIEF DESCRIPTION OF THE DRAWINGS

The embodiments herein may be better understood by referring to thefollowing description in conjunction with the accompanying drawings inwhich like reference numerals indicate identically or functionallysimilar elements, of which:

FIG. 1 illustrates an example network of devices shown with variouscommunication and powering configurations;

FIG. 2A illustrates an example electric distribution system;

FIG. 2B illustrates an example poly-phase electric distribution if thesystem shown in FIG. 2A;

FIGS. 3A-B illustrate example phase representations of the poly-phaseelectric distribution system;

FIG. 4 illustrates an example computation/measurement device;

FIG. 5 illustrates an alternative example computation/measurementdevice;

FIG. 6 illustrates an example implementation of phasor determination;

FIG. 7 illustrates another example implementation of phasordetermination, particularly for balanced waveforms;

FIG. 8 illustrates another example implementation of phasordetermination, particularly where complex coefficient notch filters areavailable; and

FIGS. 9A-9B illustrate an example simplified procedure for instantaneousphasor determination in a poly-phase electric system.

DESCRIPTION OF EXAMPLE EMBODIMENTS Overview

According to one or more embodiments of the disclosure, threesubstantially simultaneous phase waveforms may be converted into a firstquadrature signal and a zero sequence signal (e.g., by a Clarketransformer). For each phase waveform, a power system digital frequencymay be determined (e.g., by an instantaneous frequency evaluator)through analysis of the first quadrature signal (e.g., in sequence withat least one additional prior quadrature signal) while eliminatingwaveform phase angles from the analysis. Subsequently, demodulation(e.g., by a complex frequency-adaptive synchrodyne demodulator) of thefirst quadrature signal and zero sequence signal based on the powersystem digital frequency results in a positive sequence phasor, anegative sequence phasor, and a zero sequence phasor.

Description

FIG. 1 is a schematic block diagram of an example simplified network 100of devices illustratively comprising various communicating andnon-communicating devices. For example, power-lines 160 may bringelectrical grid power from respective transformers 170 intohomes/businesses/etc. to power one or more end-devices 115, generallyvia a meter 150. In addition, “pole-top” routers 120, such as field arearouters (FARs) may communicate data packets 140 (e.g., traffic and/ormessages) with other communicating nodes/devices of the network 100. Forinstance, the links between the devices may be wired links (e.g., forpower-line communication “PLC” and/or ethernet) or may comprise awireless communication medium. An energy controller (e.g., home energycontroller, “HEC”) 110 or other energy controller may be present atcertain locations, and may be in communication with the meters 150,pole-top routers 120, or else directly to another computer network,e.g., WAN 130, similar to conventional computers 105. In addition, acentralized control center or management center 190 may be present inthe network 100, such as at an electrical grid company's centralizedlocation, and may be in communication over power-lines 160 and throughWAN 130.

Smart object networks, in particular, are a specific type of networkhaving spatially distributed autonomous devices such as sensors,actuators, etc. For example, sensor networks, such as for “Smart Grid”and “Smart Cities” (e.g., for Advanced Metering Infrastructure or “AMI”applications), may cooperatively monitor physical or environmentalconditions at different locations, such as, e.g., energy/powerconsumption, resource consumption, etc. Another type of smart objectincludes actuators, e.g., responsible for turning on/off an engine orperform any other actions. Generally, smart object networks may includeany type of device that is able to communicate information on a computernetwork, such as household appliances (air conditioners, refrigerators,lights, etc.), industrial devices (heating, ventilating, and airconditioning (HVAC), pumps, motors, etc.), and other “smart” devices.Smart object networks are typically interconnected by a communicationnetwork, such as a wireless network, though wired connections are alsoavailable, and may often consist of wireless nodes in communicationwithin a field area network (FAN). For instance, each smart device(node) in a smart object network may generally be equipped with a radiotransceiver or other communication port, a microcontroller, and anenergy source, such as a battery (or, in particular to the embodimentsherein, a distribution grid power source). Typically, size and costconstraints on sensor nodes result in corresponding constraints onresources such as energy, memory, computational power and bandwidth.

Notably, though an illustrative embodiment herein relates to smartobject networks, the techniques described below may also be used by asingle device and/or a non-communicating device, such as a power gridmeasurement device (e.g., mentioned below). Also, those skilled in theart will understand that any number of nodes, devices, links, etc., aswell as any different (and suitable) type of nodes, devices, links,etc., may be present in the network, and that the view shown herein isfor simplicity and is not meant to limit the scope of the embodimentsherein. In fact, those skilled in the art will appreciate that countlessarrangements of power grid components and communicating devices may beestablished.

As noted above, electric power is generally transmitted from generationplants to end consumers (industries, commercial, residential, etc.) viaa transmission and distribution grid consisting of a network of powerstations and substations interconnected by transmission circuits/powerlines. From the transmission grid, power may then be distributed to endconsumers via a distribution system. Once at the end consumers,electricity can be used to power any number of devices, such asend-devices 115.

FIG. 2A illustrates a vastly simplified view of an example electricpower transmission and distribution grid 200 to the example devices ofFIG. 1, above. For instance, a distribution center 210 supplieselectricity over a plurality of power lines 160 to the devices atlocations “A” through “J”.

The transfer of alternating-current (AC) electric power to the end usersmost frequently takes the form of poly-phase electric power, where,e.g., three voltage waveforms are produced that are generally equal inmagnitude and 120° out of phase to each other. Each phase may generallybe used to power entire buildings, neighborhoods, etc., and may alsosupply power to many (e.g., tens, hundreds, thousands) of devices withinthose establishments. For smaller customers (e.g., households) usually asingle phase is taken to the property. For larger installations(commercial buildings and industrial facilities), all three phases maybe taken to a distribution panel, from which both single and multi (two-or three-phase) circuits may be fed. As shown in FIG. 2B, therefore,electrical power of three phases, L1, L2, and L3, is supplied to thelocations A-J (a neutral/ground may be shared by the phases).

As further noted above, the basic mathematical object that describes anAC power system waveform (current of voltage) is the “phasor” (phaseangle vector). The mathematical basis for phasors was established bySteinmetz in 1893 and has been used in electrical engineering eversince. Phasors represent constant frequency sinusoids as vectors ineither polar (magnitude and phase angle) or complex (real and imaginary)form. It is possible and common to perform AC circuit calculations usingphasors. In three-phase systems, for instance, phasors necessarily comein threes, and may be balanced (all three phases have equal magnitudeand the inter-phasor angles are uniform at 120 degrees each), orunbalanced (not all amplitudes are equal and/or inter-phasor angles arenot all 120 degrees).

FIG. 3A illustrates an example phase representation 300 of thedistribution grid's electrical power. In particular, three waveforms areillustratively produced (L1, L2, and L3) that are generally equal inmagnitude and approximately 120° out of phase to each other. Thecurrents returning from the end users to the supply transformer allshare the neutral wire (neutral point 305). If the loads are evenlydistributed on all three phases, as they are in FIG. 3A, the sum of thereturning currents in the neutral wire is zero. Any unbalanced phaseloading such as in FIG. 3B, however, may result in a current 306 at theneutral point (e.g., a harmonic distortion in the current), which maycause inefficient use of transformers, or other problems, including (butnot limited to) brown-outs or black-outs in extreme cases. (Note thatgenerally, overload of the neutral is a more frequent occurrence,resulting in heating of the neutral, since normally, the substationrelay on that circuit should detect negative sequence or zero sequenceover-current and trip long before any impact on generators is felt.)There are many factors that may create imbalance between the phases,such as excess load usage, downed power lines, etc.

Notably, it is well known that one can convert unbalanced three-phasephasors into three sets of balanced phasors (a type of superposition).The three balanced phasor sets are known as the positive sequence, thenegative sequence, and the zero sequence. This decomposition is oftendone for a variety of reasons and many power protective devices aredesigned to use one or more of the sequences to perform their functions.The concept and method of symmetric components was invented by Fortescuein 1918; it became widely accepted in the power engineering field afterWWII.

In 1988, devices that calculate phasors from power waveforms weredeveloped by Phadke and Thorp at Virginia Tech and have since beencommercialized by several companies. Such devices are known as PhasorMeasurement Units (PMUs). Because phase angle is a relative quantity, itis generally necessary when combining phasors taken from different partsof a power grid to align the phase angle elements to a common phasereference; this is typically done in PMUs through the use of GPS timingsignals. Such phasors are known as synchrophasors. PMUs measuresynchrophasors, but existing devices use phasor calculation methods thathave a number of shortcomings. These shortcomings include, among otherthings:

-   -   1. PMUs generally only calculate the positive sequence, even for        unbalanced phasors, but all three sequences are needed for many        applications.    -   2. PMUs use batch-type calculations to produce phasor values,        and so are somewhat limited by the need to accumulate batches of        waveform samples; most PMUs produce 30 or 60 phasors per second        in North America (25 or 50 in Europe), whereas the industry        wishes to go to 120 phasors per second or more (e.g., 240/second        and even 7200/second). The more advanced protection schemes are        sub-cycle in nature, which is one instance where the faster        phasor sample rate requirements arise.    -   3. Synchrophasor calculation methods introduce various errors:        -   a. Phase lag and group delay due to the use of internal high            order filters.        -   b. Additional group delay due to the need to buffer a set of            waveform samples before the phasor calculation (typically            involving a Discrete Fourier Transform).        -   c. Synchrophasor correctness depends on the frequency for            which the phasors are calculated vs. actual power system            frequency but power system frequency changes dynamically due            to changing load and generation conditions, so calculated            phasors can be in error if system frequency is assumed to be            constant and known. Various compensation techniques exist,            but they all suffer from inability to use instantaneous            frequency, and most methods for determining instantaneous            frequency involve either batch calculations, feedback (such            as in phase locked loop approaches), iterative calculations            (some sinusoid parameter trackers), or poor transient            dynamics.    -   4. Existing PMUs are expensive and are mostly deployed in        transmission substations, even though there are significant        applications for phasor measurement on distribution grids.    -   5. The computations necessary to generate synchrophasors from        waveform samples are burdensome to embedded processors.        Consequently, the deployment of phasor measurement, especially        on distribution grids, is limited.

Though the topology of the electric distribution grid typicallyconsiders the approximate balancing of the three-phase system, and otherstabilization factors of the distribution grid in general, variousfactors, such as physical events and/or the dynamic nature of energyutilization in general, may result in imbalance and/or destabilizationof the grid. At the present time, the uses for synchrophasor measurementinclude:

-   -   1. Real time monitoring and control—early indication of grid        problems, instability, inter-area oscillation, voltage        instability (e.g., operator decision support).    -   2. State determination—based on greater measurement and less        estimation; boundary state for Regional Transmission        Organization (RTO)/Independent System Operator (ISO)        applications; WAMS (Wide Area Measurement System, or        alternatively, Wide Area Management Services).    -   3. Regional transmission congestion management—to operate the        grid according to true dynamic limits.    -   4. Post-disturbance analysis—orders of magnitude time savings in        diagnosing events; also expedites power restoration.    -   5. Benchmarking system models, validation, fine-tuning—improved        parameter values support better flow models.    -   6. Power system restoration—reduced risk of unsuccessful        reclosings.    -   7. Protection and Control for distributed generation—precise        islanding, microgrid operations, synchronization.    -   8. Multi-terminal transmission line protection.    -   9. Directional and distance relaying; fault impedance relaying.    -   10. Overload monitoring and dynamic rating (real time line        impedance).    -   11. Adaptive protection—improve relay algorithms by making them        adjust to real time conditions.    -   12. Real time automated control—automated prevention of angular        and voltage stability problems, reduced low frequency        oscillations (modal power oscillation damping); nonlinear        flexible AC transmission systems (FACTS) control for grid        stability (static VAR compensators or “SVCs,” static        compensators or “STATCOMs,” Dynamic Voltage Restorers or “DVRs,”        United Power Flow converters or “UPFCs,” etc.).    -   13. System integrity protection schemes—early and accurate        determination of when power system is headed into instability.    -   14. Distribution level stabilization via distribution static        compensators (DSTATCOMs).    -   15. Distribution level pre-fault analysis.    -   16. Distribution level fault detection/classification/location.    -   17. Dynamic power grid asset rating and utilization        optimization.

Other applications will continue to be developed as PMU data becomesavailable to system operators and distribution engineers. Note that forprotection applications, as mentioned above, it is desirable to act on asub-cycle basis, so the availability of per-sample period phasors is akey enabler of advanced protection schemes. Though as noted, standardapproaches to synchrophasor determination use batch calculations andonly produce the positive sequence at rates of 30-60 reports per second.

Instantaneous Phasor Determination

The techniques herein convert three-phase voltage or current signals toall three symmetric components on an instantaneous basis, directlyincorporating instantaneous power system frequency. In particular, thetechniques herein determine symmetric components and unbalancedsynchrophasors from three-phase power line voltage or current signals.In addition, the techniques calculate instantaneous phasors and useinstantaneous power system frequency in the demodulation process so thatphasors errors do not result from mismatch of assumed frequency andactual power system frequency or power system frequency dynamics.Moreover, as described herein, a simplification is available that can beautomatically used when the inputs represent balanced phasors.

Specifically, according to one or more embodiments of the disclosure asdescribed in greater detail below, three substantially simultaneousphase waveforms may be converted into a first quadrature signal and azero sequence signal (e.g., by a Clarke transformer). For each phasewaveform, a power system digital frequency may be determined (e.g., byan instantaneous frequency evaluator) through analysis of the firstquadrature signal (e.g., in sequence with at least one additional priorquadrature signal) while eliminating waveform phase angles from theanalysis. Subsequently, demodulation (e.g., by a complexfrequency-adaptive synchrodyne demodulator) of the first quadraturesignal and zero sequence signal based on the power system digitalfrequency results in a positive sequence phasor, a negative sequencephasor, and a zero sequence phasor.

Illustratively, the techniques described herein may be performed byhardware, software, and/or firmware. FIG. 4 is a schematic block diagramof an example device 400 that may be used with one or more embodimentsdescribed herein, e.g., as an appropriately configuredmeasurement/computation device, such as a head-end device (orapplication) within the central management center 190, a FAR 120, ameter 150, an energy controller 110, a PMU, etc. The device 400 maycomprise, as one simple computer-implemented representation, a networkinterface 410, a processor 420, and a memory 440 interconnected by asystem bus 450. Notably, the device may also be powered by a powersupply 460 attached to the power grid (power-line 160).

The network interface 410 contains the mechanical, electrical, andsignaling circuitry for communicating data over physical and/or wirelesslinks coupled to the network 100. The network interface may beconfigured to transmit and/or receive data using a variety of differentcommunication protocols, including, inter alia, various wired orwireless protocols, powerline communication (PLC) protocols, broadbandover power lines (BPL), etc.

The memory 440 comprises a plurality of storage locations that areaddressable by the processor 420 for storing software programs and datastructures associated with the embodiments described herein. Theprocessor 420 may comprise necessary elements or logic adapted toexecute the software programs and manipulate the data structures 445. Anoperating system 442, portions of which are typically resident in memory440 and executed by the processor, functionally organizes the device by,inter alia, invoking operations in support of software processes and/orservices executing on the device. These software processes and/orservices may comprise an illustrative “phasor determination” process448, for use as described herein, as well as other processes not shownfor clarity.

Phasor determination process 448 may contain computer executableinstructions executed by the processor 420 to perform functions relatingto the novel techniques described herein. It will be apparent to thoseskilled in the art that other processor and memory types, includingvarious computer-readable media, may be used to store and executeprogram instructions pertaining to the techniques described herein.Also, while the description illustrates various processes, it isexpressly contemplated that various processes may be embodied as modulesconfigured to operate in accordance with the techniques herein (e.g.,according to the functionality of a similar process).

Moreover, while in certain embodiments the techniques herein may beimplemented as a software process (e.g., process 448), additional oralternative embodiments may be implemented as hardware, software,firmware, or a combination thereof. For example, FIG. 5 illustrates asimplified logical/hardware model of a phasor measurement/computingdevice 500 that may be operated to perform one or more techniquesdescribed herein (e.g., a logical representation of software/modules ofphasor determination process 248 and/or hardware/firmware componentsconfigured to operate accordingly). For example, as described in greaterdetail below, a waveform input 510 may be supplied to a Clarketransformer 520, which, based on a balance detector 530, may supply aresult to a frequency evaluator 540, which then provides input to ademodulator 550. Illustratively, the demodulator may comprise a positivesequence phasor portion 552, negative sequence phasor portion 554, andzero sequence phasor portion 556, as well as a plurality of notchfilters 560, each of which being described below, to produce phasors(phasor symmetric components) 570. Optionally, as also mentioned below,the phasors 570 produced as a result of the demodulation may be passedthrough an inverse Fortescue transformer 580 to produce an unbalancedphasor set 590.

Operationally, the Clarke transformer 520 converts three substantiallysimultaneous phase waveforms (input 510) into a quadrature signal and azero sequence signal. The Clarke transform is known in the art toconvert three-phase signals into a quadrature signal and a zero sequencesignal, and has been used mainly in AC motor control theory. Also, aninstantaneous frequency evaluation process (frequency evaluator 540)then analyzes the sequence of Clarke transformer quadrature outputs todetermine power system digital frequency on each waveform sample, whileeliminating input waveform phase angles from consideration. Theanalysis, as shown below, is based on a sequence of the quadraturesignal from the Clarke transformer and at least one additional priorquadrature signal. Additionally, a complex frequency-adaptivesynchrodyne demodulator 550 with three sub-sections (552, 554, and 556)is configured to demodulate, based on the power system digitalfrequency, the quadrature and zero-sequence signals into positive,negative, and zero sequence phasors 570. Note that the demodulator alsoincludes a set of notch filters 560, e.g., to remove double frequencysignal components that arise in the demodulation process (shown below).

Worth noting, is that the synchrodyne was invented around 1932 as anincremental improvement to the homodyne demodulator. The homodyne wasdeveloped as a radio communications device, whereas the synchrodyne wasdeveloped as a measurement instrument. Both involve a local oscillatoroperating at the same frequency as the signal carrier. The techniquesherein illustratively utilize a complex digital implementation (wherecomplex means using real and imaginary or in-phase and quadraturecomponents).

In accordance with an illustrative embodiment, the techniques alsoinclude an optional inverse Fortescue transformer 580 to convert thesymmetrical components phasors 570 into an unbalanced phasor set 590 ifdesired. As mentioned above, Fortescue invented the concept of symmetriccomponents representation of unbalanced phasors. The Fortescue transform(not used herein) converts unbalanced phasors to equivalent symmetriccomponents (positive, negative and zero sequences), while the inversetransform (used herein) converts form symmetric components to theequivalent unbalanced phasors.

Note that elimination of harmonic components of the input signals iswell-known, and is thus not discussed herein.

In accordance with one or more embodiments of the techniques herein, afirst example implementation is now detailed with reference to FIG. 6.As shown in FIG. 6, each set of three-phase signal samples a_(n), b_(n),and c_(n), (acquired simultaneously, or close enough so no significantskew error is generated) is processed via Clarke transform 520 togenerate one quadrature signal (s=α+j β) and one zero sequence signal(V₀). As shown in FIG. 6, the signal s is a sum of two scaled complexexponentials with the form:

s=e ^(jωt) K ₁ +e ^(−jωt) K ₂*

where K₁ is the positive sequence phasor and K₂* is the complexconjugate of the negative sequence phasor. Through the frequencyevaluator 540, the signal s (s_(n)) and two past quadrature signalvalues (s_(n-1) and s_(n-2)) are used to calculate the negative of thesquare of the instantaneous digital system frequency θ. Theinstantaneous digital frequency is converted to real/imaginary form andis used to update the quadrature oscillator of the demodulator 550 byone clock tick. The digital frequency may also be converted to analogsystem angular frequency ω as an output for use by other applications.The frequency evaluation takes advantage of the form of the complexsignal s. The ratio of second derivative of s to the present value ofsignal s yields the negative of the square of digital frequency. Inpractice, the second difference and the present signal value may be usedto obtain θ.

The quadrature oscillator may be phase adjusted by pulses from a GPS(global positioning satellite) timing source. In particular, by addingthe GPS timing, the techniques herein provide for synchrophasormeasurement in addition to merely phasor measurement, i.e.,synchronizing the phasors as regards phase angle. (Phase angle is arelative quantity so a common basis is needed, otherwise phasorsmeasured at different points on a distributed circuit may not becombined arithmetically.) GPS is one illustrative example to allow forglobal referencing, though other references can be used local to thepoint of measure, i.e., not in combination with any phasors measured byany other unsynchronized measurement means. Notably, the oscillatorphase may be reset once per second, or on a more frequent basis, such asonce per nominal power cycle or an integer multiple thereof.

The K₁ and K₂* demodulators work by using complex arithmetic todownshift or upshift the signal s, as shown in FIG. 6. The upshift(portion 552) of the quadrature signal moves the negative sequencemodulation frequency to zero and doubles the positive sequencefrequency. Likewise, the downshift (portion 554) of the quadraturesignal sets the positive sequence modulation frequency to zero anddoubles the negative sequence frequency. The positive sequence phasorand negative sequence phasor may then be correspondingly obtainedthrough removal of a double frequency component resulting from theupshifting and downshifting, respectively. That is, the two-channelnotch filters 560 with illustrative filter configuration as shown inFIG. 6, remove the double frequency components with asymptotically zerogroup delay and phase lag, leaving the DC or near DC components, whichare the complex phasor coefficients desired (K₁ and K₂). Note that forthe negative sequence, it is necessary to perform a complex conjugate toobtain the complex phasor coefficient K₂.

The zero sequence demodulation works in generally the same way, exceptthat we need only one demodulation that uses one of either the positivefrequency or a negative frequency of the zero sequence signal, since thephasor coefficient K₃ is the same on both the positive and negativefrequencies:

V ₀ =e ^(jωt) K ₃ +e ^(−jωt) K ₃*

As mentioned above, the phasor coefficients K₁, K₂, and K₃ may be inputto an inverse Fortescue transformer 580 (e.g., with an illustrativevalue for q) to produce an optional unbalanced phasor output 590.

Note that while the embodiment shown in FIG. 6 uses an illustrativetechnique to evaluate instantaneous frequency, other methods ofdetermining digital frequency might be used. The method disclosed hereinhas the advantage of handling dynamic variations of system frequencyautomatically. It also has the disadvantage, however, of being adifferential method, and therefore could be noisy under some conditions.Smoothing of the digital frequency could be employed in thesecircumstances but is not desirable, as this would introduce dynamic laginto the process.

Also note that the notch filters shown in the unbalanced phasor processof FIG. 6 are illustratively second order recursive, and are designedfrom standard notch filter prototypes. However, standard prototypes donot provide sufficiently narrow reject bands, and have group delay andphase lag near DC that may be considered too large for the techniquesherein. This may be resolved by modifying the prototype through arecursive feedback stabilization process during filter synthesis thatprovides a new “sharply defined” second order filter with very high Q(sharpness) at the desired doubled frequency, and therefore a narrowreject band and near zero group delay and phase lag at DC. Moreover, tomake the filters adaptive to changes in power system frequency, theprocess can modify the filter coefficients on each waveform sampleperiod. Therefore the filters will track the system frequencyvariations. While in one embodiment, the filter coefficients may beupdated on each sample, an alternative embodiment herein provides thatthe filter coefficients be updated less often, e.g., depending uponsystem frequency dynamics. In fact, the filter coefficients may beupdated based on automatically determining when system frequencydeviates by more than a threshold amount from the last value of systemfrequency used to update the filter coefficients.

For the case where the input signals represent balanced phasors, thatis, determining that the three substantially simultaneous phasewaveforms are balanced, the techniques herein may be simplifiedsignificantly and automatically. In particular, the zero sequence signalfrom the Clarke transformer (V₀) is exactly zero for a balanced phasorset. By thresholding a rectified version of V₀ with a small amount ofoptional low order smoothing (first or second order recursive), thesystem can detect the balanced case and switch to a simplified technique(set of steps) as illustrated in FIG. 7.

In the simplified process shown in FIG. 7, frequency evaluation may bebased on only present and one prior value for signal s, which ispossible due to the simplified form of s. In this instance, thedemodulator 550, in response to determining that the three substantiallysimultaneous phase waveforms are balanced (imbalance indicator), needonly determine the positive sequence phasor K₁ (the negative and zerosequence phasors are exactly zero). No notch filters are required herein FIG. 7 because no frequency doubling occurs in the demodulation stepin the balanced case. Note that FIG. 7 also illustrates an exampletechnique for determining whether the phasors are balanced or not byprocessing the V₀ output of the Clarke transformer (i.e., based on thezero sequence signal). In practice, this step would be taken beforeperforming any of the frequency evaluation or demodulation steps.

In accordance with one or more additional or alternative embodimentsherein, the solution of FIG. 6 (and FIG. 7 in the balanced case) can beimproved in the following way: the six notch filters (three 2-channelfilters) can be replaced with two notch filters that have complexcoefficients plus one complex subtraction (subtractor). This additionalversion of the technique is detailed in FIG. 8, and has the advantage ofhaving fewer filters and thus less opportunity for phase distortion. Inparticular, this model may be implemented in software or logic (e.g., anASIC), and is thus not limited to real coefficients. That is, since thepositive and negative sequence phasors are differentiated by theircomplex components, real filters (as in FIG. 6) are unable to make thatdifferentiation when notching out particular frequencies. By usingcomplex notching, however, the negative frequency may be removed fromthe positive sequence phasor K₁. To then obtain the negative sequencephasor K₂, half of the signal s (s_(a)) may be subtracted from s (by thesubtractor 858) to extract the other half of the signal (s_(b)) forinput to the downshifting portion 554.

Said differently, FIG. 8 illustrates an alternative embodiment where afirst complex coefficient notch filter (560) is utilized/implementedbetween an output of the Clarke transformer and an input of thedemodulator for the positive sequence phasor (552), and a second complexcoefficient notch filter (560) is used between the output of the Clarketransformer and an input of the demodulator for the zero sequence phasor(556). The complex subtractor 858 may then be utilized/implementedbetween the output of the Clarke transformer and an input of thedemodulator for the negative sequence phasor to subtract an output ofthe first complex coefficient notch filter (s_(a)) from the output ofthe transformer (s) for the input (s_(b)) of the demodulator for thenegative sequence phasor (554). The remainder of the operation issimilar to that shown and described for FIG. 6 above.

FIGS. 9A-9B illustrate an example simplified and generic procedure forinstantaneous phasor determination in a poly-phase electric system inaccordance with one or more embodiments described above, from theperspective of a computing/measuring device 200 and/or 500 (or somecombination thereof). The procedure 900 starts at step 905, andcontinues to step 910, where, as described in greater detail above,three substantially simultaneous phase waveforms 510 are converted intoa first quadrature signal and a zero sequence signal (e.g., by Clarketransformer 520).

If the zero sequence signal indicates that the signal is unbalanced instep 915, then (according to FIG. 6 or FIG. 8) the procedure continues(along the solid line) to step 920 to determine the power system digitalfrequency for each phase waveform through analysis of a sequence of thefirst quadrature signal and two or more prior quadrature signals whileeliminating waveform phase angles. Accordingly, in step 925, the process(e.g., demodulator 550) demodulates, based on the power system digitalfrequency, the first quadrature signal and zero sequence signal intopositive, negative, and zero sequence phasors 570. For example, asdescribed above, step 925 may correspondingly upshift, downshift, removedouble frequencies by applying notch filters 560 (e.g., complex notches,adaptive notches), etc. Also, optionally, in step 930 the procedure 900may convert the symmetrical components of the positive, negative, andzero sequence phasors into an unbalanced phasor set 590 (e.g., via aninverse Fortescue transformer 580).

Returning to step 915, in the event the zero sequence signal indicatesthat the signal is balanced, then (according to FIG. 7) the procedurecontinues (along the dashed line) to step 935 to determine the powersystem digital frequency for each phase waveform through analysis of asequence of the first quadrature signal and a single prior quadraturesignal, while eliminating waveform phase angles. Then, in step 940, thedemodulator 550, based still on the power system digital frequency,demodulates the first quadrature signal into the positive sequencephasor (again, the negative sequence phasor and zero sequence phasor arezero).

Note that as an option to both balanced and unbalanced waveforms, theprocedure 900 may convert the power system digital frequency into ananalog system angular frequency output in step 945. The procedureillustratively ends in step 950, though may continue to update phasorvalues based on additional samples, accordingly. It should be noted thatwhile certain steps within procedure 900 may be optional as describedabove, the steps shown in FIGS. 9A-9B are merely examples forillustration, and certain other steps may be included or excluded asdesired. Further, while a particular order of the steps is shown, thisordering is merely illustrative, and any suitable arrangement of thesteps may be utilized without departing from the scope of theembodiments herein. Moreover, while the procedure 900 is described as asingle procedure, certain steps from the procedure may be incorporatedinto other procedures, and the procedure shown is not meant to beall-inclusive.

The novel techniques described herein, therefore, provide forinstantaneous phasor determination in a poly-phase electric system. Inparticular, the novel techniques convert digitized three-phase ACvoltage or current signals representing unbalanced phasors into fullsymmetric component phasors on a per waveform sample basis (e.g., anoutput on every waveform sample without phase lag), while taking intoaccount instantaneous three-phase power frequency. Specifically, thetechniques herein have the characteristic of calculating the phasorswithout the use of high order filters (especially long non-recursivefilters), feedback loops, phase-locked loops, or iterative processingtechniques, and avoid any need to buffer any more than the immediate setof waveform samples, so no storage of past waveform samples is needed.Moreover, the process does not require any post-computation correctionfor frequency errors, and has a significant simplification in the casewhere the waveforms are in fact balanced, providing a fast method fordetermining if the balance exists before calculating what would turn outto be unnecessary negative and zero sequence phasor coefficients.

Furthermore, the techniques according to one or more embodiments abovehave the additional benefit of comprising a lightweight set ofcomputations as compared to standard phasor calculation methods, so thatimplementation in devices such as field area routers (FARs) andconnected grid routers (CGRs) or other embedded processors is efficientand compact from code size, code execution, and data storagestandpoints. That is, the techniques herein may be used with smart gridtechnologies as part of an integrated smart sensor strategy, forexample, allowing each substation and field router to act as a smartsensor, thus eliminating the need for external smart sensor remoteterminal units (RTUs). For instance, waveform transducers may beconnected via analog-to-digital conversion cards to a router, and thetechniques herein may thus be implemented in the router itself. Electricutilities could thus use the techniques herein to increase gridobservability, and the synchrophasor components described herein couldbe communicated to grid analytics and control applications that mightreside locally, at a substation, or at a control center. Illustratively,for example, FARs and/or CGRs with this capability may be used ondistribution circuits and at Points of Common Coupling for microgridsand Distributed Energy Resources to obtain key grid state information atany location where such devices are deployed.

While there have been shown and described illustrative embodiments thatprovide for instantaneous phasor determination in a poly-phase electricsystem, it is to be understood that various other adaptations andmodifications may be made within the spirit and scope of the embodimentsherein. For example, while the embodiments above generally describe thepoly-phase source system as a three-phase system, this is merely oneexample embodiment of a poly-phase system (granted, the most prevalenttype today), and is not meant to limit the embodiments herein. Also, theembodiments have been shown and described herein with relation toparticular computational components, such as Clarke transformers,inverse Fortescue transformers, notch filters, etc., and moreparticularly, to certain implementations (formulas, constants,multipliers, etc.) However, the embodiments in their broader sense arenot as limited, and may, in fact, be used with other types of suitablecomputation components and/or implementations. In addition, as notedabove, while the techniques above may have made specific reference totransmission systems or distribution systems, the disclosure hereinapplies to both the transmission and distribution portions of theelectric grid, where applicable.

The foregoing description has been directed to specific embodiments. Itwill be apparent, however, that other variations and modifications maybe made to the described embodiments, with the attainment of some or allof their advantages. For instance, it is expressly contemplated that thecomponents and/or elements described herein can be implemented assoftware being stored on a tangible (non-transitory) computer-readablemedium (e.g., disks/CDs/etc.) having program instructions executing on acomputer, hardware, firmware, or a combination thereof. Accordingly thisdescription is to be taken only by way of example and not to otherwiselimit the scope of the embodiments herein. Therefore, it is the objectof the appended claims to cover all such variations and modifications ascome within the true spirit and scope of the embodiments herein.

1. A computer-implemented method, comprising: converting threesubstantially simultaneous phase waveforms into a first quadraturesignal and a zero sequence signal; determining a power system digitalfrequency for each phase waveform through analysis of the firstquadrature signal while eliminating waveform phase angles from theanalysis; and demodulating, based on the power system digital frequency,the first quadrature signal and zero sequence signal into a positivesequence phasor, a negative sequence phasor, and a zero sequence phasor.2. The method as in claim 1, further comprising: converting symmetricalcomponents of the positive sequence phasor, negative sequence phasor,and zero sequence phasor into an unbalanced phasor set.
 3. The method asin claim 1, wherein determining the power system digital frequency isthrough analysis of the sequence of the first quadrature signal and twoadditional prior quadrature signals.
 4. The method as in claim 1,further comprising: converting the power system digital frequency intoan analog system angular frequency output.
 5. The method as in claim 1,wherein demodulating comprises: obtaining the positive sequence phasorby i) upshifting the first quadrature signal to move a negative sequencemodulation frequency to zero and ii) removing a first double frequencycomponent resulting from the upshifting to obtain the positive sequencephasor; and obtaining the negative sequence phasor by i) downshiftingthe first quadrature signal to move a positive sequence modulationfrequency to zero and ii) removing a second double frequency componentresulting from the downshifting to obtain the negative sequence phasor.6. The method as in claim 5, wherein demodulating further comprises:obtaining the zero sequence phasor through a single demodulation thatuses one of either a positive frequency or a negative frequency of thezero sequence signal.
 7. The method as in claim 5, wherein removing thefirst and second double frequency components comprises: utilizing aplurality of notch filters to remove the first and second doublefrequency components.
 8. The method as in claim 7, wherein the pluralityof notch filters are synthesized based on recursive feedbackstabilization to provide sharply defined second order filtering.
 9. Themethod as in claim 7, further comprising: adapting one or more filtercoefficients of the plurality of notch filters in response to afrequency deviation greater than a threshold between the power systemdigital frequency for the first quadrature signal and the at least oneadditional prior quadrature signal.
 10. The method as in claim 1,further comprising: determining that the three substantiallysimultaneous phase waveforms are balanced; and wherein demodulatingcomprises, in response to determining that the three substantiallysimultaneous phase waveforms are balanced, demodulating, based on thepower system digital frequency, the first quadrature signal into thepositive sequence phasor only, wherein the negative sequence phasor andzero sequence phasor are exactly zero.
 11. The method as in claim 10,wherein determining that the three substantially simultaneous phasewaveforms are balanced comprises determining the balance based on thezero sequence signal prior to determining the power system digitalfrequency and demodulating.
 12. The method as in claim 1, whereindetermining the power system digital frequency is through analysis ofthe sequence of the first quadrature signal and at least one priorquadrature signal.
 13. The method as in claim 1, further comprising:utilizing a first complex coefficient notch filter between an output ofthe transformer and an input of the demodulator for the positivesequence phasor; utilizing a second complex coefficient notch filterbetween the output of the transformer and an input of the demodulatorfor the zero sequence phasor; and utilizing a complex subtractor betweenthe output of the transformer and an input of the demodulator for thenegative sequence phasor to subtract an output of the first complexcoefficient notch filter from the output of the transformer for theinput of the demodulator for the negative sequence phasor.
 14. Themethod as in claim 1, wherein the positive sequence phasor, negativesequence phasor, and zero sequence phasor are synchrophasors.
 15. Anapparatus, comprising: a Clarke transformer to convert threesubstantially simultaneous phase waveforms into a first quadraturesignal and a zero sequence signal; an instantaneous frequency evaluatorto determine a power system digital frequency for each phase waveformthrough analysis of the first quadrature signal while eliminatingwaveform phase angles from the analysis; and a complexfrequency-adaptive synchrodyne demodulator to demodulate, based on thepower system digital frequency, the first quadrature signal and zerosequence signal into a positive sequence phasor, a negative sequencephasor, and a zero sequence phasor.
 16. The apparatus as in claim 15,further comprising: an inverse Fortescue transformer to convertsymmetrical components of the positive sequence phasor, negativesequence phasor, and zero sequence phasor into an unbalanced phasor set.17. The apparatus as in claim 15, wherein the demodulator comprises: afirst portion to obtain the positive sequence phasor by i) an upshift ofthe first quadrature signal to move a negative sequence modulationfrequency to zero and ii) a removal of a first double frequencycomponent resulting from the upshift to obtain the positive sequencephasor; a second portion to obtain the negative sequence phasor by i) adownshift of the first quadrature signal to move a positive sequencemodulation frequency to zero and ii) a removal a second double frequencycomponent resulting from the downshifting to obtain the negativesequence phasor; and a third portion to obtain the zero sequence phasorthrough a single demodulation that uses one of either a positivefrequency or a negative frequency of the zero sequence signal.
 18. Theapparatus as in claim 17, further comprising: a plurality of notchfilters to remove the first and second double frequency components. 19.The apparatus as in claim 15, wherein the demodulator is furtherconfigured to, in response to a determination that the threesubstantially simultaneous phase waveforms are balanced: demodulate,based on the power system digital frequency, the first quadrature signalinto the positive sequence phasor only, wherein the negative sequencephasor and the zero sequence phasor are exactly zero.
 20. The apparatusas in claim 15, wherein determining the power system digital frequencyis through analysis of the sequence of the first quadrature signal andat least one prior quadrature signal.
 21. The apparatus as in claim 15,further comprising: a first complex coefficient notch filter between anoutput of the transformer and an input of the demodulator for thepositive sequence phasor; a second complex coefficient notch filterbetween the output of the transformer and an input of the demodulatorfor the zero sequence phasor; and a complex subtractor between theoutput of the transformer and an input of the demodulator for thenegative sequence phasor to subtract an output of the first complexcoefficient notch filter from the output of the transformer for theinput of the demodulator for the negative sequence phasor.
 22. Atangible, non-transitory, computer-readable media having softwareencoded thereon, the software, when executed by a processor, operableto: convert three substantially simultaneous phase waveforms into afirst quadrature signal and a zero sequence signal; determine a powersystem digital frequency for each phase waveform through analysis of thefirst quadrature signal while eliminating waveform phase angles from theanalysis; and demodulate, based on the power system digital frequency,the first quadrature signal and zero sequence signal into a positivesequence phasor, a negative sequence phasor, and a zero sequence phasor.23. The computer-readable media as in claim 22, wherein the softwarewhen executed is further operable to: convert symmetrical components ofthe positive sequence phasor, negative sequence phasor, and zerosequence phasor into an unbalanced phasor set.
 24. The computer-readablemedia as in claim 21, wherein the software when executed is furtheroperable to: determine that the three substantially simultaneous phasewaveforms are balanced based on the zero sequence signal prior todetermining the power system digital frequency and demodulating; and inresponse: demodulate, based on the power system digital frequency, thefirst quadrature signal into the positive sequence phasor only, whereinthe negative sequence phasor and the zero sequence phasor are exactlyzero.
 25. The computer-readable media as in claim 21, wherein thesoftware when executed is further operable to: implement a first complexcoefficient notch filter between an output of the transformer and aninput of the demodulator for the positive sequence phasor; implement asecond complex coefficient notch filter between the output of thetransformer and an input of the demodulator for the zero sequencephasor; and implement a complex subtractor between the output of thetransformer and an input of the demodulator for the negative sequencephasor to subtract an output of the first complex coefficient notchfilter from the output of the transformer for the input of thedemodulator for the negative sequence phasor.